I want to prove the following algebraically (that is, NOT with calculus): If an object with initial velocity is accelerated at some constant acceleration over a time , and finishes with a final velocity of , then the average velocity of the object is . This average velocity is the equivalent velocity that this object would have to be travelling (without acceleration) to travel the same distance as the object experiencing acceleration as above.
I need to somehow prove this in order to derive the kinematic equations. But how? It is easy to derive the kinematic equations with a knowledge of calculus, but if you are teaching this to someone without a knowledge of calculus, how can this be proven rigorously?